Finite Group Databasebeta

← #10828783

Subgroups — #10828783

Order 768.

1164 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#553C211yesmaximal, characteristic
2C2×#5033C211yesmaximal, characteristic, complemented
3#82373C211yesmaximal, characteristic, complemented
4C22×#808C211yesmaximal, characteristic, complemented
5#71739C212yesmaximal, complemented
6#71681C211yesmaximal, characteristic, complemented
7#71680C212yesmaximal, complemented
8#71610C211yesmaximal, characteristic, complemented
9#71607C212yesmaximal, complemented
10#71576C212yesmaximal, complemented
11#71554C211yesmaximal, characteristic, complemented
12#24091—31nomaximal, Sylow
13C3×#553—24no—
14C3×#553—24no—
15C2×C3×#126—21no—
16C2×C3×#125C2211yescharacteristic
17C2×C3×#123—21no—
18C2×C3×#121—21no—
19C2×C3×#121C2211yescharacteristic
20C2×C3×#121C2211yescharacteristic
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117C2211yescharacteristic
25C22×C3×C4⋊C4C2211yescharacteristic
26C2×#935—21no—
27C22×C4⋊Dic3C2211yescharacteristic
28C23×C3⋊C8C2211yescharacteristic
29C2×#832C2212yes—
30C2×#832C2212yes—
31#5212C2212yes—
32#5212C2211yescharacteristic
33C2×#830C2212yes—
34C2×#830C2211yescharacteristic
35#5033—24no—
36#4981—21no—
37C2×#808—24no—
38C2×#808C2212yes—
39C2×#808C2212yes—
40C2×#807C2212yes—
41#4953—21no—
42#4943—21no—
43#4928C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#553D311yescharacteristic
49C2×#480—31no—
50C22×#106—31no—
51#2486—31no—
52#2059—32no—
53#2034—32no—
54#1893—31no—
55#1869—31no—
56#1864—32no—
57#1854—32no—
58#1824—31no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×Dic3—21no—
68C3×#126—28no—
69C3×#125—22no—
70C3×#125—22no—
71C3×#123—28no—
72C3×#121—28no—
73C3×#121—22no—
74C3×#121—22no—
75C3×#121—22no—
76C3×#121D414yes—
77C3×C4×D4—28no—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×#117—22no—
83C3×#117—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D414yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C42—21no—
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—22no—
103#935—44no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—22no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2312yes—
108C2×C4⋊Dic3C2312yes—
109C2×Dic3⋊C4—22no—
110C22×C3⋊C8—22no—
111C22×C3⋊C8—24no—
112C22×C3⋊C8C2312yes—
113C22×C3⋊C8C2312yes—
114#832—24no—
115#832—24no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118C3⋊#95—21no—
119#830—22no—
120#830—24no—
121#808—24no—
122#808—24no—
123#807—24no—
124C23×C8—31no—
125#553—24no—
126#553—24no—
127C2×#126—21no—
128C2×#125C2×D311yescharacteristic
129C2×#123—21no—
130C2×#123—61no—
131C2×#121—21no—
132C2×#121C2×D311yescharacteristic
133C2×#121C2×D311yescharacteristic
134C2×C4×D4—21no—
135C2×C4×D4C2×D311yescharacteristic
136C2×C4×D4C2×D311yescharacteristic
137C2×#117C2×D311yescharacteristic
138C22×C4⋊C4—31no—
139C22×C4⋊C4C2×D311yescharacteristic
140#480—64no—
141C2×#107—32no—
142C2×#106—32no—
143C2×#106—32no—
144C2×#106—64no—
145C2×#102—32no—
146C2×#102—32no—
147C2×C22⋊C8—31no—
148C2×C22⋊C8—32no—
149#398—61no—
150#397—61no—
151#389—32no—
152#381—61no—
153#380—31no—
154#380—32no—
155#340—32no—
156#340—32no—
157#340—32no—
158#336—32no—
159C24×C3—21no—
160C24×C3C2×D411yescharacteristic
161C24×C3C2×D411yescharacteristic
162C2×C3×D4—22no—
163C2×C3×D4—24no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—24no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C2×C3×D4—82no—
173C2×C3×D4C2×D412yes—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—24no—
180C22×C3×C4—24no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—82no—
186C22×C3×C4—82no—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—42no—
197C22×Dic3—42no—
198C22×Dic3—42no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4C2×D412yes—
207C3×C4⋊C4C2×D412yes—
208C3×C4⋊C4C2×D412yes—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—22no—
215C3×C42—24no—
216C3×C42—21no—
217C3×C42—21no—
218#217—44no—
219#217—42no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—22no—
223C4⋊Dic3—22no—
224C4⋊Dic3—42no—
225Dic3⋊C4—44no—
226C2×C3⋊C8—28no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8#4114yes—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4#6711yescharacteristic
236C22×D4C22×D311yescharacteristic
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—61no—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D311yescharacteristic
242C22×C8—32no—
243C22×C8—32no—
244C22×C8—64no—
245C22×C8—62no—
246#126—28no—
247#125—22no—
248#125—22no—
249#123—28no—
250#123—124no—
251#121—28no—
252#121—22no—
253#121—22no—
254#121—22no—
255#121#6714yes—
256C4×D4—28no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—22no—
261#117—22no—
262#117—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4#6711yescharacteristic
274C2×C4⋊C4#6714yes—
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4#6711yescharacteristic
285C2×C22⋊C4#6711yescharacteristic
286C2×C42—21no—
287C2×C42C22×D311yescharacteristic
288#107—64no—
289#106—64no—
290#106—64no—
291#102—64no—
292#102—64no—
293C22⋊C8—64no—
294C22⋊C8—62no—
295#95—62no—
296#95—61no—
297#95—62no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3—42no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3#12012yes—
311C23×C3#13512yes—
312C23×C3AGL(1,ℤ/8)12yes—
313C23×C3C22×D411yescharacteristic
314C3×D4—24no—
315C3×D4—44no—
316C3×D4—48no—
317C3×D4—41no—
318C3×D4—41no—
319C3×D4—84no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—84no—
342C2×C3×C4—84no—
343C2×C3×C4#12012yes—
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C2×#4112yes—
347C2×C3×C42−1+412yes—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—84no—
355C3⋊C8—24no—
356C3⋊C8—42no—
357C24—21no—
358C24C2×#6711yescharacteristic
359C24C2×#6711yescharacteristic
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—82no—
371C2×D4C2×#6712yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—122no—
388C22×C4—122no—
389C22×C4—122no—
390C22×C4C2×#6711yescharacteristic
391C22×C4C2×#6711yescharacteristic
392C22×C4C2×#6711yescharacteristic
393C22×C4C2×#6711yescharacteristic
394C22×C4C2×#6712yes—
395C22×C4C2×#6712yes—
396C22×C4C23×D311yescharacteristic
397C2×C8—34no—
398C2×C8—68no—
399C2×C8—62no—
400C2×C8—62no—
401C2×C8—62no—
402C2×C8—62no—
403C2×C8—62no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—62no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C4⋊C4—62no—
415C4⋊C4—124no—
416C4⋊C4—122no—
417C4⋊C4C2×#6712yes—
418C4⋊C4C2×#6712yes—
419C4⋊C4C2×#6712yes—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—22no—
424C22⋊C4—24no—
425C22⋊C4—22no—
426C22⋊C4—124no—
427C22⋊C4—122no—
428C42—24no—
429C42—21no—
430C42—21no—
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—41no—
438C22×C3—42no—
439C22×C3—41no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—41no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3#44914yes—
446C22×C3C2×#12011yescharacteristic
447C22×C3#54712yes—
448C22×C3C2×#13511yescharacteristic
449C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
450C22×C3#57712yes—
451C3×C4—22no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—42no—
455C3×C4—41no—
456C3×C4—42no—
457C3×C4—42no—
458C3×C4#52511yescharacteristic
459C3×C4#53611yescharacteristic
460C3×C4#54712yes—
461Dic3—82no—
462Dic3—82no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23—22no—
469C23—21no—
470C23—22no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23#94912yes—
476C23#95012yes—
477C23#95312yes—
478C23C22×#6711yescharacteristic
479D4—24no—
480D4—44no—
481D4—48no—
482D4—41no—
483D4—41no—
484D4—84no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—244no—
515C2×C4#95312yes—
516C2×C4D3×#4112yes—
517C2×C4#101012yes—
518C2×C4C22×#6711yescharacteristic
519C2×C4C22×#6711yescharacteristic
520C8—64no—
521C8—122no—
522C2×C3—24no—
523C2×C3—81no—
524C2×C3—82no—
525C2×C3#240611yescharacteristic
526C2×C3C2×#44911yescharacteristic
527C2×C3#300911yescharacteristic
528C2×C3#333712yes—
529C2×C3#334512yes—
530C22—22no—
531C22—22no—
532C22—24no—
533C22—24no—
534C22—41no—
535C22—42no—
536C22—41no—
537C22—42no—
538C22—42no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—82no—
544C22#523814yes—
545C22C2×#94911yescharacteristic
546C22C2×#95011yescharacteristic
547C22#582412yes—
548C22#583612yes—
549C22C2×#95311yescharacteristic
550C4—22no—
551C4—41no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—42no—
557C4—242no—
558C4—242no—
559C4#582412yes—
560C4#582511yescharacteristic
561C4#582811yescharacteristic
562C3#2409111yescharacteristic, complemented, Sylow
563C2—24no—
564C2—81no—
565C2—82no—
566C2#8360612yes—
567C2#8360812yes—
568C2C2×#523811yescharacteristic
569C2#8409011yescharacteristic
570C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#553C2
socleC23×C3C22×D4
hypercenterC2×#553D3
hypocenterC3#24091
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#5033C2semidirect
3. #82373C2semidirect
4. C22×#808C2semidirect
5. #71739C2semidirect
6. #71681C2semidirect
7. #71680C2semidirect
8. #71610C2semidirect
9. #71607C2semidirect
10. #71576C2semidirect
11. #71554C2semidirect
562. C3#24091semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#5532
C2×#50332
#823732
C22×#8082
#717392
#716812
#716802
#716102
#716072
#715762
#715542
#240913